# 2-cocycle for a Lie ring action

## Definition

Suppose $L$ is a Lie ring, $M$ is an abelian group, and $\varphi:L \to \operatorname{End}(M)$ is a Lie ring homomorphism from $L$ to the ring of endomorphisms of $M$, where the latter gets the usual Lie ring structure from its structure as an associative ring.

### Explicit definition

A 2-cocycle for this action is a homomorphism of groups $f:L \times L \to M$ (where $L \times L$ is the external direct product) satisfying the additional condition:

$\! \varphi(x)(f(y,z)) - \varphi(y)(f(x,z)) + \varphi(z)(f(x,y)) - f([x,y],z) - f(y,[x,z]) + f(x,[y,z]) = 0\forall \ x,y,z \in L$

If we suppress $\varphi$ and denote the action by $\cdot$, this can be written as:

$\! x \cdot f(y,z) - y \cdot f(x,z) + z \cdot f(x,y) - f([x,y],z) - f(y,[x,z]) + f(x,[y,z]) = 0\forall \ x,y,z \in L$

### Definition in terms of the general definition of cocycle

A 2-cocycle for a Lie ring action is a special case of a cocycle for a Lie ring action, namely $n = 2$.

### Group structure

The set of 2-cocycles for the action of $L$ on $M$ forms a group under pointwise addition.

## Particular cases and variations

Case or variation Condition for an additive homomorphism to be a 2-cocycle Further information
$M = L$ and $\varphi$ is the adjoint action, i.e., $\varphi(x)$ is the inner derivation induced by $x$ $\! [x,f(y,z)] - [y,f(x,z)] - [z,f(x,y)] - f([x,y],z) - f(y,[x,z]) + f(x,[y,z]) = 0\forall \ x,y,z \in L$
$\varphi$ is the zero map $\! f(x,[y,z]) = f([x,y],z) + f(y,[x,z]) \ \forall \ x,y,z \in L$ 2-cocycle for trivial Lie ring action
$L$ is an abelian Lie ring $\! y \cdot f(x,z) = x \cdot f(y,z) + z \cdot f(x,y) \ \forall \ x,y,z \in L$
$\varphi$ is the zero map and $L$ is an abelian Lie ring No additional condition